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(a) Determine the interval(s) on which the function is increasing. Determine the interval(s) on which the function is decreasing. 

To determine the intervals where the function is increasing or decreasing, we can analyze the graph you provided. Here's the step-by-step process:

### Increasing Intervals:
A function is increasing where the graph goes upward as you move from left to right.

1. **From \( x = -3 \) to \( x = -1 \):** The graph moves upward from the point \((-3, 2)\) to \((-1, -2)\).
2. **From \( x = 1 \) to \( x = 2 \):** The graph moves upward from the point \((1, -3)\) to \((2, 4)\).

So, the function is increasing on the intervals:
\[
\text{Increasing: } (-3, -1) \cup (1, 2)
\]

### Decreasing Intervals:
A function is decreasing where the graph goes downward as you move from left to right.

1. **From \( x = -4 \) to \( x = -3 \):** The graph moves downward from the point \((-4, 4)\) to \((-3, 2)\).
2. **From \( x = -1 \) to \( x = 1 \):** The graph moves downward from the point \((-1, -2)\) to \((1, -3)\).
3. **From \( x = 2 \) to \( x = 3 \):** The graph moves downward from the point \((2, 4)\) to \((3, 2)\).

So, the function is decreasing on the intervals:
\[
\text{Decreasing: } (-4, -3) \cup (-1, 1) \cup (2, 3)
\]

### Summary:
- **Increasing:** \( (-3, -1) \cup (1, 2) \)
- **Decreasing:** \( (-4, -3) \cup (-1, 1) \cup (2, 3) \)

Would you like a detailed explanation of why these intervals are classified as increasing or decreasing, or do you have any specific questions?

### Relative Questions:
1. What are the critical points of this function?
2. How can you determine the maximum and minimum values from the graph?
3. What is the significance of the intervals where the function is neither increasing nor decreasing?
4. How would the intervals change if the graph were shifted up or down?
5. How would you calculate the average rate of change over a specific interval?

**Tip:** Understanding how to identify increasing and decreasing intervals on a graph is crucial for analyzing the behavior of functions, particularly in calculus.

Determine the interval(s) on which the function is increasing. Determine the interval(s) on which the function is decreasing.

Understand how to determine the intervals where a function is increasing or decreasing by analyzing the graph. This problem covers key calculus concepts, including the first derivative test and graph analysis. Suitable for students in Grades 10-12.

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